A fresh snack from one of two vending machines

Question

Machine A keeps its snacks fresh 70\% of the time; the other 30\% are stale. Machine B is worse: only 40\% fresh, 60\% stale. You pick a machine at random (a fair coin decides) and buy one snack. It turns out to be fresh. What is the probability you used Machine A?

Accepted Answer

Each machine starts equally likely. Picture 100 attempts, about 50 at each machine: - Machine A: 50 	imes 0.70 = 35 fresh snacks. - Machine B: 50 	imes 0.40 = 20 fresh snacks. Of the 35 + 20 = 55 fresh snacks, 35 came from Machine A: P(	ext{A} \mid 	ext{fresh}) = \frac{35}{55} = \frac{7}{11} \approx 0.636. Getting a fresh snack is more consistent with Machine A, so your belief rises from 50 percent to about 64 percent. Takeaway. When two sources could explain what you observed, weight each by how often it produces that observation, then normalize.

A fresh snack from one of two vending machines

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